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1 Chapter 29: Atomic Structure What will we learn in this chapter? Contents: Electrons in atoms Wave functions Electron spin Pauli exclusion principle Atomic structure Periodic table W. Pauli & N. Bohr Both electron spin and the Pauli exclusion principle are the missing ingredients for the understanding of the periodic table of elements.

2 Electrons in atoms: wave functions Last class: When treating electrons in atomistic length scales we are unable to determine both position and momentum precisely. We need to introduce a new description in term of probabilities. Definition: Wave function Similarly to classical strings, describe electrons by wave functions. A wave function Ψ( r 1, r 2,..., r n ) describes n electrons with space coordinates r i =(x i,y i,z i ). The state of the n-electron system is fully described by Ψ. The wave function corresponds to the probability density to find the particles at a given point in space. Think of it as a cloud. If, for example, the particles have charge, the wave function describes also the charge distribution of the particles. Wave functions contd. The following quantities can be computed from a wave function: Average position of a particle Average velocity of a particle Dynamic quantities (momentum, energy, angular momentum). Example: spherically-symmetric states of the hydrogen atom probabilities densities (projection) recall Bohr s orbits

3 Selected states of the hydrogen atom How are wave functions determined? Schrödinger equation (1926): i 2 Ψ( r, t) = t 2m 2 Ψ( r, t)+v ( r, t)ψ( r, t) The corresponds to a second-order derivative in the space coordinate. The physical information of the system is fully described by the potential V between the particles. If V is known, the properties of the system can, in principle, be computed. The solutions of the Schrödinger equation represent various possible states of the system. For many systems solutions only exists for given energy levels, i.e., quantization is introduced (e.g., angular momentum and energy).

4 Quantization of the angular momentum Bohr s model: The angular momentum quantization was done ad hoc. Schrödinger equation: The angular momentum quantization is a necessary condition for the existence of solutions. Permitted values of angular momentum: The possible values for L of an electron with energy En and quantum number n in a hydrogen atom are L = l(l + 1) L = L l =0, 1, 2,... n 1 The component of L in a given direction (e.g., z component) can only have the set of values L z = m l m l =0, ±1, ±2,..., ±l The magnitude of ml cannot exceed l. Hydrogen atom angular momentum Within the full quantum-mechanical description, the hydrogen atom is described now by four quantum numbers: n, l, ml, s: Principal quantum number:!!!!! n Angular momentum quantum number:!! Magnetic quantum number:!!!!! ml Electron spin (later ):!!!!!! s l For each energy level En, there are several distinct states with the same energy. The magnetic quantum number describes small shifts in the energy levels (and thus in the wavelengths) when the atom is placed in a magnetic field. The component Lz can never be quite as large as L.

5 Angular momentum While we can determine precisely L and Lz, the component of L perpendicular to the z axis cannot be determined and is confined to a cone of direction (Pauli exclusion between L components). Electron spin In the 1920s it was noticed that energy levels would split even when no external field was present. This suggests that the electron has an intrinsic angular momentum, in addition to the angular momentum associated with the orbital motion. Picture electrons as charged spinning spheres. The spin is also quantized. For the quantization along the z-axis ( ) S S = = S = S z = ± 1 = s 2 Electrons are also called spin-1/2 particles (or Fermions). Simple analogy (solar system): Both earth and sun spin around their axes and earth spins around the sun. The total angular momentum is the vector sum of these.

6 Cross section of selected states n determines the energy, l and ml the angular momentum: n = 1 n = 2 l = 0 l = 1 n = 3 l = 1 l = 2 Many-electron atoms So far: We have only studied the simplest atom with one electron, H. When an atom has more electrons, electron-electron interactions need to be included. Although the Schrödinger equation can describe these systems, analytical solutions do not even exist for Helium (He)! Central field approximation: Assume that one electron feels the average field of the other electrons and the nucleus. In this case the quantum numbers of the angular momentum are the same as for the hydrogen atom. The energy levels now depend on n and l with the the usual restrictions on (n, l, ml, s).

7 Pauli exclusion principle To implement the central field approximation, we need to take into account exclusion principles. Naive approach: One would expect that in the ground state of an atom all electrons are in the same lowest state. If this was the case then the behavior of atoms with increasing number of electrons should show gradual changes in physical and chemical properties. Reality proves this naive approach wrong: Fluorine (9 electrons) is a halogen and very reactive. Neon (10 electrons) is an inert gas (no compounds). Sodium (11 electrons) is an alkali metal. Pauli exclusion principle contd. The key to the change in the behavior of atoms when the number of electrons is increased was discovered by Wolfgang Pauli (1925). Pauli exclusion principle: No two electrons in an atom can occupy the same quantum-mechanical state. Alternatively, no two electrons in an atom can have the same values of all four of their quantum numbers. We can now generate a list of all possible sets of quantum numbers and thus possible states of the electrons in an atom. Customary notation: the value of l is designated by a letter: l = 0! l = 1! l = 2! s state p state d state l = 3! f state!!! followed by g,

8 First four shells of electron quantum states factor 2 due to spin X-ray levels States with the same n form a shell, with the same l a subshell. Atomic structure Structure of a neutral atom: The atom has Z electrons, Z protons and a number of neutrons in the core. Z is called the atomic number. Neutrons have no charge. Quantum regions closes to the nucleus have lowest energies. The outer electrons determine the behavior of the atom. Building an atom: Start with a bare nucleus with Z protons. Successively add electrons paying attention to the exclusion rules.

11 The periodic table of elements Proceeding with the atom building, we can understand the regularities seen in the periodic table of elements for each column. A slight problem occurs with the M and N shells because the 3d and 4s subshells overlap in energy: Theses elements have 1 or 2 electrons in the 4s state and increasing numbers in the 3d states. These elements are all transition metals for Z = 21 to 30. Similarly, for Z = 57 to 71 two electrons are in the 6s level but the 4f and 5d levels are only partially filled. These are the rare earth elements. At Z = 91 another such series starts with the actinides. X-ray energy levels Outer electrons of an atom: Responsible for optical spectra. Emit in the near-visible region with energies between 2 3 ev. Vacancies in the inner shells of complex atoms: Responsible for x-ray energy levels. Because these electrons are close to the nucleus, binding energies are larger and more energy is required to remove these. Suppose an electron is knocked out of the K shell. It can be filled with an electron of the L, M, N, shells emitting a photon. If the outermost electrons are in the N shell, the x-ray spectrum has 3 lines from transitions from the L, M, N to the K shell. K series:

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